Positive and negative exact boundary controllability results for the linear Biharmonic Schr\"odinger equation

TL;DR

This paper establishes conditions under which the linear Biharmonic Schrödinger equation with hinged boundary conditions is exactly controllable via boundary control, highlighting the role of the parameter racd0 in controllability.

Contribution

It provides a precise characterization of boundary controllability for the linear Biharmonic Schrödinger equation, identifying critical parameter values affecting controllability.

Findings

Controllability holds for all racd0<0 except a countable set.

Spectral analysis and nonharmonic Fourier series are used in the proof.

The controllability time T can be arbitrarily small.

Abstract

In this paper, we study the exact boundary controllability of the linear Biharmonic Schr\"odinger equation $i\partial_ty=-\partial_x^4y+ \gamma\partial_x^2y$ on a bounded domain with hinged boundary conditions and boundary control acts on the second spatial derivative at the {left} endpoint, where the parameter $\gamma<0$. We prove that this system is exactly controllable in time $T>0$, if and only if, the parameter $\gamma$ does not belong to a critical countable set of negative real numbers. The analysis in this work is based on spectral analysis together with the nonharmonic Fourier series method.